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Every Lipschitz continuous map is uniformly continuous, and hence continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli theorem implies that if { f n } is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.
Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1). A contraction mapping has at most one fixed point.
Conversely, any such sequence (f k) of Lipschitz functions converges to an α –Hölder continuous uniform limit f. Any α –Hölder function f on a subset X of a normed space E admits a uniformly continuous extension to the whole space, which is Hölder continuous with the same constant C and the same exponent α.
The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable (the size of a function domain interval over which function value differences are less than ) that depends on only , while in (ordinary) continuity there is a locally applicable that depends on both and . So uniform ...
Met is not the only category whose objects are metric spaces; others include the category of uniformly continuous functions, the category of Lipschitz functions and the category of quasi-Lipschitz mappings. The metric maps are both uniformly continuous and Lipschitz, with Lipschitz constant at most one.
For uniform continuity, δ may depend on ε and ƒ. For pointwise equicontinuity, δ may depend on ε and x 0. For uniform equicontinuity, δ may depend only on ε. More generally, when X is a topological space, a set F of functions from X to Y is said to be equicontinuous at x if for every ε > 0, x has a neighborhood U x such that
The limit function is also Lipschitz continuous with the same value K for the Lipschitz constant. A slight refinement is A slight refinement is A set F of functions f on [ a , b ] that is uniformly bounded and satisfies a Hölder condition of order α , 0 < α ≤ 1 , with a fixed constant M ,
That is, a function is Lipschitz continuous if there is a constant K such that the inequality ((), ()) (,) holds for any ,. [15] The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.